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Mirrors > Home > MPE Home > Th. List > ralimdaa | Structured version Visualization version GIF version |
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.) |
Ref | Expression |
---|---|
ralimdaa.1 | ⊢ Ⅎ𝑥𝜑 |
ralimdaa.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ralimdaa | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimdaa.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralimdaa.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) | |
3 | 2 | ex 449 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
4 | 1, 3 | ralrimi 2940 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
5 | ralim 2932 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 df-ral 2901 |
This theorem is referenced by: eltsk2g 9452 ptcnplem 21234 poimirlem26 32605 allbutfifvre 38742 climleltrp 38743 fnlimabslt 38746 stoweidlem61 38954 stoweid 38956 fourierdlem73 39072 smflimlem2 39658 |
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